Let's analyze the functions and equations from the image. The function provided is:

$f(x) = 2x + 3$

You are asked to evaluate the function at specific inputs:

1. $f\left(\frac{3}{2}\right)$
2. $f(a+h)$
3. $f(a) + f(h)$

Let's solve each one:

1. $f\left(\frac{3}{2}\right)$: $f\left(\frac{3}{2}\right) = 2\left(\frac{3}{2}\right) + 3 = 3 + 3 = 6$

2. $f(a+h)$: $f(a+h) = 2(a+h) + 3 = 2a + 2h + 3$

3. $f(a) + f(h)$: $f(a) = 2a + 3 \quad \text{and} \quad f(h) = 2h + 3$ $f(a) + f(h) = (2a + 3) + (2h + 3) = 2a + 2h + 6$

Summary:

1. $f\left(\frac{3}{2}\right) = 6$
2. $f(a+h) = 2a + 2h + 3$
3. $f(a) + f(h) = 2a + 2h + 6$

Would you like more details or have any further questions?

Here are some related questions to explore further:

1. What is the general formula for evaluating $f(x) = mx + b$ at different values of $x$?
2. How does the result of $f(a+h)$ compare with $f(a) + f(h)$ for linear functions?
3. What would $f(a-h)$ equal for the given function?
4. How would you find the inverse function of $f(x) = 2x + 3$?
5. What would happen if the function were quadratic instead of linear?

Tip: For linear functions of the form $f(x) = mx + b$, the function's behavior is entirely determined by the slope $m$ and the intercept $b$.